Problem: All of the 3rd grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$3.00$ each for students, and the group paid $$37.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$15.00$ each for teachers and $$12.50$ each for students, and the group paid $$107.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+3y = 37.5}$ ${15x+12.5y = 107.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-15x-6y = -75}$ ${15x+12.5y = 107.5}$ Add the top and bottom equations together. $ 6.5y = 32.5 $ $ y = \dfrac{32.5}{6.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {7.5x+3y = 37.5}$ to find $x$ ${7.5x + 3}{(5)}{= 37.5}$ $7.5x+15 = 37.5$ $7.5x = 22.5$ $x = \dfrac{22.5}{7.5}$ ${x = 3}$ You can also plug ${y = 5}$ into $ {15x+12.5y = 107.5}$ and get the same answer for $x$ ${15x + 12.5}{(5)}{= 107.5}$ ${x = 3}$ There were $3$ teachers and $5$ students on the field trips.